Point and vector positioning, using location determination (LD) systems such as GPS, provide measures of heights above a reference ellipsoid, such as the WGS84 ellipsoid, as a perpendicular distance from the ellipsoid surface. However, surveyors are interested in elevation, which is measured by perpendicular distance from the geoid (a proxy for sea level) above the local datum defined by the surveyor's local control points, which may be location coordinates for locations that have already been surveyed, used for consistency checks.
A body rotating with the Earth experiences the gravitational attraction of the Earth (and other celestial bodies) and a centrifugal force generated by the Earth's rotation about the Earth axis. This produces a gravitational force vector g(x,y,z), which depends upon location coordinates (x,y,z), which may be described as a sum of gravitational field and a centrifugal potential field. This sum defines a set of equipotential surfaces W(x,y,z)=constant on which the magnitude of the vector g is constant. Each of these equipotential surfaces is known as a geoid and is discussed in A. Leick, GPS Satellite Surveying, John Wiley & Sons, New York, Second Edition, 1995, pp. 215-232. Because the local gravitational attraction will differ for a location near a mountain range and a location with no topographic relief, a geoid surface is not smooth everywhere and has bumps or undulations.
Current real-time kinematic Global Positioning Systems (GPS) allow the user to determine, in a least-squares calculation, a set of parameters that relate measured or keyed-in WGS84 heights above the ellipsoid to local control elevations (benchmarks). In effect, this approach models the relationship between the local vertical datum and the ellipsoid as an inclined plane with parameters that describe the location and orientation of the best fitting plane.
To determine the precise range from an Earth-based location determination receiver to a satellite, a reference coordinate system is chosen such that the instantaneous location of the satellite and the receiver are expressed in a uniform coordinate system. The Global Positioning System (GPS) utilizes a Cartesian, Earth-centered, Earth-fixed, coordinate system for determining this range. In the GPS system, the positive axis points in the direction of 0.degree. longitude, the positive y-axis points in the direction of 90.degree. East longitude, and the xy-plane defines the Earth's equatorial plane. To transform Cartesian coordinates into the latitude, longitude and height coordinates of the receiver, a physical model of the Earth is adopted. This model is based on an oblate ellipsoid having a semimajor axis length a and a semiminor axis length b, with b.ltoreq.a. The values for the lengths a and b are chosen to match as closely as possible a mean sea level or geoid surface. One such closely matching ellipsoid is the WGS84 ellipsoid, which has a semimajor axis length a=6378.137 km and a flatness factor f=1-(b/a)=1/298.257223563 (Leick, op cit, p. 487). Other closely matching ellipsoids include the NAD27, WGS72 and NAD83 ellipsoids, each with its own ellipsoid parameters. In some instances, a "local" ellipsoid that better matches a local region is used in place of the WGS84 or other global ellipsoid.
Local or global geodetic coordinates are sufficient to define horizontal network coordinates. However, vertical coordinates are traditionally referenced relative to a geoid rather than to an ellipsoid, which by definition has a smooth shape. The shape of the selected geoid, however, is influenced by the mass distribution in the Earth, and by the resulting local gravity gradient or variation. In geographic regions where the distribution of mass is homogeneous and the gravity variation is negligible, the difference between the geoid surface and the ellipsoid surface may be adequately represented by a vertical offset, normal to the ellipsoid surface. In regions where the gravity variation is non-negligible but constant, the difference between the geoid surface and the ellipsoid surface is better represented by a selected vertical offset and selected tilt angles along two orthogonal axes (vertical plane adjustment). However, in regions where the distribution of the Earth's mass is non-homogeneous or where survey measurements are performed over large spatial distances, large fluctuation in the gravity gradients can occur, and the planar model relating height relative to the geoid and the ellipsoid degrades in accuracy. For example, on the plains of Kansas, a planar model might be sufficient for a 100.times.100 km (kilometer) project area, whereas at the foot of the Rocky Mountains a planar model may provide only a good approximation on a 3.times.3 km.sup.2 project area, as indicated in FIG. 1. Assuming a tilted plane relationship between the geoid and ellipsoid for zones such as 1, 2, and 3 in FIG. 1 is often sufficient. However, if a survey project spans an entire mountain front, this simple approach is insufficient.
By incorporating a geoid model in real time processing, geoid undulations with large wavelengths (in a range of about 3-100 km, depending on the quality of the model) might be accommodated in a local site calibration. This is particularly important for users working in regions where the geoid shape departs from an ellipsoidal shape over short spatial distances, and for users who need long range real time kinematic (RTK) capability in surveying and mapping products.
What is needed is an approach that combines a geoid model with a vertical plane adjustment and that allows for derivation of orthometric heights from WGS84 or similarly measured or keyed-in locations on small or large project regions in the field in real time. Preferably, the resulting orthometric heights should be accurate to within about 30 cm.